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This section is about the computational machinery in a neural circuit.
The synapses that occur along the circuit is represented by an ordered
sequence. The term "ordered" means that the firing of a synapse
depends on the state of the neural circuit before the synapse occurred.
The transistion into each new state of the system depends on the previous
states [1].
The neuron is describe by a state function of the form
x' = U (x,q)
where x and q are variables representing sequences of ordered events.
State transition equation can explicitly represent the propagation
or recurrence of a state in consequence observable steps.
x(k+1) = U (k;k+1) x(k)
x(k) is the kth state, and U (k;k+1) is the transition to the
k+1 state. This is the recurrence relationship of an ordered
movement of energy flow.
|
|
n0 | x00 --> x01 --> x02 --> x03 --> x04
n | U0(01) U0(12) U0(23) U0(34)
n1 | x10 --> x11 --> x12 --> x13 --> x14
n | U1(01) U1(12) U1(23) U1(34)
e n2 | x20 --> x21 --> x22 --> x23
u | U2(01) U2(12) U2(23)
r n3 | x30 --> x31 --> x32 --> x33
o | U3(01) U3(12) U3(23)
n |
s nn | xn0 --> xn1 --> xn2 --> xn3 --> xn4
| Un(01) Un(12) Un(23) Un(34)
|_______________________________________
0
state(t)
Sequence diagram for n+1 cell automata
In the synthetic neuron model, groups of synchronous neurons in a
circuit is grown by making connections between a groups of neurons
in a circuit. The cluster of neurons in a circuit is compose of sets
or lists of artificial neurons. The figure above shows the synapses
of the n-th neuron which can be written as a list of state transitions
x_n (i).
The Language of Modelling Neurons
The lisp language was created with simple, yet elegant, programming
methods for computing lists efficiently. One of these methods
is recursion. To create recursive code, we try to break down
the process into simpler subprocesses. The process of breaking down
a complex set of sequences into a set of simpler sequences of instructions
on a computer is called reduction. The process of reduction is analogous
to the process of clustering used in pattern recognition. As clustering
is an aggregative concept which enables the recognition of more complex
epicentric paradigms, reduction is a process of simplifying complex
sequences by breaking up these long sequences into shorter simpler ones.
Lisp tries to efficiently process recursive loops by using a technique
called continuations in which the code branches back into itself like
using a "goto" statement. This alleviates the necessity of repeatedly
branching out of the code by calling a subroutine.
Reduction is a way of transforming a long hard problem into many
short simple ones. Reduction as used in studying complex problems
has some strong parallels with the term clustering which is used to
reduce complexity in pattern recognition. They are basically similar
paradigms. Our quest is to find intelligent ways to cluster synchronous
neural circuits together in large part by using the experience of
building the software code in the formalism of lisp.
State Propagation In Neural Networks
The state transition matrix describes the propagation from observable,
consecutive states k to k+1. U(k;k+1) contains the essential information
in the neural system.
x(k+1) = U (k;k+1) x(k)
The kernel, U, contains the code for the ordered sequence of state
transitions. The form of these transitions has the tiling structure
of a binary tree.
A sequence is an ordered binary list. The sequence below is an ordered
branch of a complete binary tree. The end point of this branch is a
sequence with leaves (daad).
Footnote
1.
Satosi Watanabe wrote [2]:
In passing, it may be noted that the (cognitive version of the) inverse
H-theorem also stems from the asymmetric interpretation of the symmetric
relation
Namely, when P(F|E) is given, p(E|F) has then to be computed by
the Bayes formula,
and the passage from the prior probability p(E)
to the ulterior probability p(E|F) results in an entropy decrease.
The fundamental rule of probability, Bayes rule, constraints the
system to be temporally ordered. The Bayes rule applied to each state
transition in a sequence of events determines the system's directional
movement in time. These state transitions are time ordered or time
dependent Markov processes because a transition probability follows
the conditional probability P(E|F) * P(F|G) * P(G|H) ..., where
event E depends on F, and F upon G, and G upon H, etc., in temporal order.
Given two state sequences X_1 and X_2 from two neurons N_1 and N_2, we
can get a measure of the cross-correlation by comparing the similarity of
the Markov transition probabilities in the sequence determined
by P(E|F) * (F|G) ... for neurons N_1 and N_2.
2.
Time and the Probabilistic View of the World, Satosi Watanabe,
The Voices of Time, ed. J. T. Fraser, 1966, p. 527
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